Octal Games on Graphs: The game 0.33 on subdivided stars and bistars

TL;DR

This paper extends octal games from heaps to graphs, analyzing the 0.33 game on subdivided stars and bistars, and provides a complete solution for these structures.

Contribution

It introduces a new framework for octal games on graphs and fully solves the 0.33 game on subdivided stars and bistars.

Findings

Complete resolution of 0.33 game on subdivided stars.

Complete resolution of 0.33 game on bistars.

Framework for analyzing octal games on various graphs.

Abstract

Octal games are a well-defined family of two-player games played on heaps of counters, in which the players remove alternately a certain number of counters from a heap, sometimes being allowed to split a heap into two nonempty heaps, until no counter can be removed anymore. We extend the definition of octal games to play them on graphs: heaps are replaced by connected components and counters by vertices. Thus, an octal game on a path P\_n is equivalent to playing the same octal game on a heap of n counters. We study one of the simplest octal games, called 0.33, in which the players can remove one vertex or two adjacent vertices without disconnecting the graph. We study this game on trees and give a complete resolution of this game on subdivided stars and bistars.